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### Keywords:

• ray tracing;
• Huygen's principle;
• ionospheric propagation;
• geometric optics

### Abstract

[1] Huygen's principle, in the form of boundary integral equations, is applied to the problem of radio wave propagation. Complex propagation is analyzed by dividing the region between transmitter and receiver into a number of zones and propagating the solution between these zones by means of integral equations with simple approximate kernels. In the limit that the regions become dense, the approach leads to a useful form of the WKB solution, and this is demonstrated through its application to the study of propagation through a disturbed ionosphere.

### 1. Introduction

[2] Huygen's principle is an appealing idea that has found application in many fields that exhibit wave propagation. The principle treats each point of a wave front as the source of a spherical wave with subsequent wave fronts formed from the envelope of these simpler waves. For most fields, the principle finds its mathematical realization in the boundary integral formulation of the governing equations. Such a formulation expresses the field at a particular point as the sum of point source solutions distributed over a surface. Unfortunately, for a nonhomogeneous medium, the requisite point source solutions can themselves be extremely complex and difficult to derive. An alternative is to use the integral formulation to propagate the solution from an initial surface to a nearby intermediate surface by means of a simple approximate point source solution. The process can then be repeated for the new surface and so on until the required propagation has been achieved. Monteath [1973] has shown that the use of a single intermediate surface, together with a geometric optics point source solution, is sufficient to solve several nontrivial electromagnetic propagation problems involving large changes in refractive index and strong diffraction. It is clear that quite complex problems can be tackled if sufficient intermediate surfaces are used. In the limit that the intermediate surfaces become dense, there essentially results a Feynman path integral representation of the fields. Linares and Moretti [1988] and Lee [1978] have studied path integral techniques for the situation where the electromagnetic propagation can be reduced to the study of the Helmholtz equation. Linares and Moretti have mainly considered the theoretical aspects of the path integral approach, but Lee has used the technique to obtain practical solutions for diffraction by multiple knife edges. More recently, Eliades [1991] and Ong and Constantinou [1996] have used the paraxial limit of the path integral method to study propagation over a plateau. It will be noted, however, that almost all practical applications of the path integral approach approximate the paths by a limited number of finite length segments. This is tantamount to the multiple application of the Helmholtz surface integral equation. In practice, the path integral approach is difficult to apply and its real utility lies in the conceptual insight that it offers.

[3] The present paper seeks to develop the multiple application of surface integral equations as an alternative technique for solving general propagation problems. This surface integral (SI) approach enables the full nonhomogeneous Maxwell equations to be studied within a framework that allows the degree of approximation to be adapted to suit the local complexity of the propagation. Section 2 develops the technique for scalar fields that satisfy a Helmholtz equation and, in section 3, the limiting form of the technique is used to develop a WKB approximation for propagation through a medium with continuously varying refractive index. The WKB approximation follows very naturally from the formulation and the equations for calculating the development of wave amplitude are of a particularly economic form. This approximate solution can be regarded as a generalization of that obtained for graded index media by Gomez-Reino and Linares [1987]. Section 4 develops the full electromagnetic form of the technique and shows how the results of section 3 can be extended to a WKB solution for the full Maxwell equations. In section 5, the WKB solution is applied to some representative propagation problems including that of propagation through a disturbed ionosphere.

### 2. Helmholtz Equation

[4] Consider a potential ψ(r) that satisfies the Helmholtz equation

over a region V and an auxiliary potential χ(r, r0) that satisfies

over the same region. Potentials χ and ψ are related by the surface integral

where S is the surface of V, r0 lies within V and ∂/∂n represents the outward normal derivative at the boundary. (Note that relation (3) holds even when the refractive index is spatially varying.) If χ is chosen such that χ = 0 on S, then

where K = −∂χ/∂n. In general, it is difficult to derive a form of χ with the requisite behavior over the whole surface S and so impractical to consider a single application of (4) as the basis for a general propagation algorithm. An alternative, however, is to use an approximate form of K to propagate the solution a small distance forward from S and then reapply (4) at the new surface. Consider a coordinate system such that propagation is essentially in the z direction. Divide the propagation region by surfaces for which z is constant and consider a K for which χ satisfies χ = 0 on these surfaces. (Although each constant z plane will need to be closed by a hemisphere at infinity, the sources of the field are assumed bounded and hence there will be no contribution to (4) from this hemisphere.) For propagation between such surfaces, the kernel will take the form

where n is the average value of refractive index between points r and r0, Δz = z − z0, Δr = , Δx = x − x0 and Δy = y − y0. This kernel, however, is only approximate and requires the surfaces to be close enough for n to be regarded as constant on the scale of Δz. In the limit that the surfaces become dense, repeated application of (4) yields what is essentially a Feynman path integral solution to the propagation problem. The multidimensional integral resulting from the repeated application of (4) can be formally reinterpreted as a sum over all possible paths that join the start and finish points of the propagation. In essence, the total propagation kernel can be calculated from

where

and D(r) is a volume element in the space of paths that join r0 to r (normalization chosen such that K(r, r0) δ(x − x0, y − y0) as z z0). In practice, however, K will be constructed as the N ∞ limit of

where Δz = and zi = z0 − iΔz. Note that, in the paraxial limit,

### 3. WKB Approximation

[5] Of great importance to electromagnetic wave calculations is the WKB approximation. This is a high frequency approximation that leads to the ray tracing techniques which lie at the heart of many propagation calculations. Consider expression (6) for the propagation kernel, then the maximum contribution to the path integral will come from paths around that for which the phase distance S is minimum (the ray path of Fermat's principle). Around this path, the lowest order variation in S will be quadratic. Ignoring variations higher than quadratic, this yields the WKB approximation [Rosen, 1969]

where Sc and δ2Sc represent S and its second variation when evaluated on the path of minimum phase. (The remaining path integral is in terms of variations q from the path of minimum phase and for which q = 0 at the end points.) Consider the path of minimum phase to be parameterized by its group path τ (dτ = ) and denote the end points (x0, z0) and (x, z) by A and B, respectively. Let r = rc(τ) be the minimum phase path expressed in three- dimensional coordinates, then this path satisfies [Jones, 1979]

Consider path variations q of the form

where the vectors Pi are unit length, perpendicular to the minimum phase path (and to each other) and satisfy

Vectors Pi are the “polarisation” vectors of the standard WKB solution to Maxwells equations [Jones, 1979, p. 594]. The second variation δ2Sc can now be expressed in the form

where (x1, x2, x3) are Cartesian coordinates in three-dimensional space. The normalization kernel

can be built up from the short path kernels

where Δτ is small, matrix Γ has coefficients γij and subscripts A and B denote quantities evaluated at the beginning and end of the short path (note that the normalization in (17) ensures a delta function characteristic as Δτ 0). Consider the path of minimum phase that joins two general points and break this into N short paths, each with group path length Δτ. Repeated application of the short path kernel will yield

where the ith path runs between points qi−1 and qi ( note that q0 = qN = 0). The above integral can be evaluated to yield

where

with

where δij is the Kronecker delta. It will be noted that both C12 and F = C11C22 are diagonal matrices and that D = Δτ2C222 + C22FC122∣. From the factorization

it follows that

where

and

Expanding the above determinant results in the difference equation

and, in the limit N ∞, the differential equation

with boundary conditions 1 and D± 0 as τ 0 (it has been assumed that n = 1 at the start of propagation).

[6] Equations (11), (13), (15) and (27) together provide a means of calculating the parameters that are required in the evaluation of the kernel

This kernel, together with integral equation (4), provides the required WKB solution to the Helmholtz equation (S(r, r0) = is the phase distance). Although the calculation of the kernel will require both P1 and P2 , it will be noted that P1, P2 and the minimum phase path are mutually orthogonal and so the solution of (13) will only be required for either P1 or P2. It will also be noted that a knowledge of P1 and P2 facilitates the calculation of a WKB solution to the full Maxwell equations for nonhomogeneous media (a fact will be exploited in the next section). At the beginning of propagation, the electric field will need to be expanded in the P1 and P2 directions. The evolution of these vectors can then be used to track the evolution of the electric field direction along a ray and the above WKB kernel its magnitude. This procedure will provide an effective approximate solution away from caustics and strong sources of diffraction.

### 4. Full Electromagnetic Equations

[7] The reciprocity theorem provides a relationship involving two otherwise unrelated time harmonic electromagnetic fields. In the present context, the important result is

which relates the value of a time harmonic electromagnetic field (E, H) at point r0 to its behavior on a surface S that separates the field sources from the point r0 . (Electromagnetic field (E0, H0) is an auxiliary field resulting from a Hertzian dipole J0 located at point r0.) This relation is extremely general and applies to nonhomogeneous media, the position dependent material properties entering through the dipole solution (E0, H0). There is a complementary relationship

for the case where field (E0, H0) results from a magnetic dipole M0 at point r0. It was shown by Monteath [1973] how the above expressions could provide an effective means of analyzing wave propagation over a lossy ground and the diffraction of waves by a metal sheet (here surface S is the vertical plane through the change in ground properties or the plane of the metal sheet). By considering (E, H) to result from a Hertzian dipole, Monteath [1973] calculated the mutual impedance between two short dipoles for the above canonical problems. Although he only used a geometric optics approximation for the dipole fields on S, Monteath [1973] was able to calculate the O(1/r2) behavior of the field at ground level and the leading order of diffraction caused by the screen. (It should be noted that the geometric optics approximation must include the effect of ground reflections if they exist.) The major contribution to the surface integrals arises from regions over which the geometric optics approximation is adequate. Consequently, the intermediate surface allows the calculation of high order effects from low order fields.

[8] It is fairly obvious that the above technique can be extended to more complex situations by introducing more than one dividing surface between the start and finish of propagation. The solution is then propagated from one surface to the next by means of an approximate dipole solution (the geometric optics approximation is normally sufficient). Such a surface integral (SI) approach will, however, require the surfaces to be placed at changes in the ground constituents or at features where significant diffraction occurs. In this way, complex transition solutions are replaced by surface integrals involving much simpler fields. The two major ingredients for this approach are a means of calculating the approximate dipole solutions and an algorithm for calculating the surface integrals. In the work of Coleman [2001], a simple free space geometric optics solution was used to calculate VHF propagation in a complex terrestrial environment. In the current paper, however, we will concentrate on applying the method to an environment with continuously varying refractive index.

[9] The Helmholtz WKB solution is extended to the full electromagnetic case by using equation (29) is instead of (3). Once again, the dominant contribution arises from the vicinity of the path of minimum phase and the solution expanded around this path. If field (E0, H0) includes the effect of an image dipole on the other side of surface S, equation (29) becomes

where n is the unit normal on the surface. If we divide up the propagation region by suitably close planes that are orthogonal to the path of minimum phase, H0n can be approximated by

where J0 is perpendicular to the propagation direction and the axes have been rotated so that z is the coordinate in the direction of propagation. (Equation (32) is essentially the paraxial limit of the tangential components of test field H0.) Consequently, components perpendicular to the propagation direction will propagate in the same manner as solutions to the Helmholtz equation. Parallel to the propagation direction, however, there is a different story. For J0 parallel to the propagation direction, and in the paraxial limit,

Consider, at some point, the electric field to be zero in the direction of propagation. Evaluating the integral in (33) by stationary phase, the increase in Ez, after distance Δz, will be

Equation (34) implies that the electric field direction rotates an amount that keeps it orthogonal to the ray path and hence is consistent with equation (13). As a consequence, the equations of section 3 fully determine the WKB solution to the full Maxwell equations. To solve for the field, we first develop the ray paths out from a source (position r0) using

We next develop the polarization vectors along a particular path using

and this allows us to track the evolution of the electric field direction. Finally, the scalar considerations of section 3 allow us to develop the magnitude and phase of the electric field along the various ray trajectories. Consider the electric field due to a bounded source at position r0. In the high frequency limit, equation (4) and kernel (28) will imply a field amplitude of the form

where heff is the effective height of the source, η is the impedance of the medium at the source and I is the current in the source feed. Furthermore, S(r, r0) = where the integral is along the ray trajectory and that quantities D± are derived from the differential equations

with boundary conditions = 1 and D± = 0 when τ = 0. (Note that quantities D± can be interpreted as the wave front principal radii of curvature.)

### 5. Some Examples

[10] We first consider the case of propagation through an ionosphere consisting of an F2 layer with plasma frequency fp given by

where h is the height above the ground, foF2 is the peak plasma frequency, hmF2 is the layer height and ymF2 is the layer thickness. The refractive index n of the ionosphere is given by n = where f is the wave frequency. The WKB solution of the previous section has been developed into a computer algorithm that can analyze propagation through the above ionosphere (and other more general distributions of refractive index). This algorithm uses a Runge-Kutta Fehlberg method [Mathews, 1987] to solve differential equations (11), (13) and (27). In the development of the computer program, it was found that great care needed to be taken when choosing the branch of the square root in equation (38). Fortunately, however, this could be achieved automatically by treating the square root as and using the branch structure of the complex square roots provided by the computer language.

[11] As a first test, we consider the WKB solution to propagation in the simple layered ionosphere characterized by equation 38 with hmF2 = 320 km, ymF2 = 70.7 km and foF2 = 12 MHz. Table 1 shows the behavior of some key quantities at the landing points of rays launched with a variety of take of angles (TOAs) and a frequency of 15 MHz. The second column shows the ground range, the third column the phase path, the fourth column shows the spreading loss (calculated from quantities D+ and D) and the final two columns show D+ and D, respectively. Included (in brackets) is the loss when evaluated by means of virtual ray deviation techniques [e.g., Nickisch, 1988; Vastberg and Lundborg, 1996]. It will be noted that the current technique produces results that are almost identical to those provided by the alternative techniques. There are, however, several advantages to the current approach. Firstly, if the polarization vectors are required, there are very few additional equations to be solved in order to obtain D+ and D. Secondly, it will be noted that the quantity D+ changes sign at the skip zone edge, a manifestation of the fact that this point lies on a caustic. Consequently, the current approach automatically produces the expected phase jump in the WKB solution as it passes through a caustic. It will be noted that the change in sign of D+ accurately locates the caustic.

Table 1. WKB Solution Parametersa
• a

Distances are in kilometers, angles are in degrees, and loss is in decibels.

524712536.817118.5(118.5)−739.32418.9
101875.11945.956118.4(118.4)−938.71876.7
1514921571.732117.4(117.4)−916.51530.5
201234.91326.484116.3(116.3)−835.61306
251053.31158.546115.3(115.3)−748.71156.9
309191039.355114.4(114.4)−666.91057.5
35816.9953.143113.6(113.6)−585.4994.5
40739.6891.742112.6(112.6)−482.2963.3
45688.7854.101110.0(110.0)−259.7972.1
46683.5850.415108.0(108.0)−163.5982.1
47681.2848.81898.3(98.1)−17.2996.9
48683.4850.326109.7(109.7)232.11019.4
49694857.312114.9(115.0)748.51055.7
50725.5877.755120.3(120.4)2401.21126.3

[12] A particularly testing example is that of round the world ionospheric propagation and for which the electromagnetic energy should focus at the antipodal point. Such propagation, however, involves multiple ground reflections and so requires reflection conditions for D± (note that the vectors Pi will satisfy the same reflection conditions as an electric field at a perfectly conducting boundary). After reflection by a spherical surface of radius re, D± will be unchanged and will jump in value by (2 + (±1 − 1)((P1 . n)2 + (P2 . n)2)) where p is a unit vector in the incident propagation direction, n is the unit normal to the surface at the point of incidence and the Pi take their incident values. Table 2 shows the behavior of some key quantities at each hop of the propagation path up until the antipodal point is reached. (The same ionosphere as above was used and a TOA of about 2.619 degrees allowed a hop to land very close to the antipodal point.) It will be noted that the zero point of D gives an accurate determination of the antipodal point (true value 20037.5).

Table 2. WKB Solution Parameters at the Ground Reflection Points Between the Ionospheric Reflectionsa
HopGround RangePhase PathLossD+D
• a

Distances are in kilometers, angles are in degrees, and loss is in decibels.

12862.82944.51117.2−475.92770.7
25725.65889.024122.7951.94992.6
38588.38833.538125.5−1427.76226
411451.111778.053126.71903.56226.5
514313.914722.567126.7−2379.54993.7
617176.717667.081124.92855.82771.8
720039.520611.59587.5−33320.4

[13] The techniques, developed in this paper, provide a full WKB solution that is able to maintain the correct phase through a caustic. In particular, it provides an effective alternative to ray deviation techniques when calculating solution amplitudes. As an example, we consider propagation through an ionosphere that is disturbed by a wave structure with

where fU is the plasma frequency of the undisturbed ionosphere (that given by equation 39), R is the ground range, ε is the strength of the disturbance, ϕ is a phase offset and λR and λh are the wavelengths of the structure. Figures 1 and 2 show the ray paths for a transmission frequency of 15 MHz and ionosphere with hmF2 = 300 km, ymF2 = 100 km, foF2 = 12 MHz for the background ionosphere and ε = 0.5, λR = 1000 km, and λh = 1000 km for the ionospheric structure (there is a phase offset of ϕ = for Figure 1 and ϕ = 0 for Figure 2). It will be noted that the ionospheric wave structure has caused a dramatic change in propagation from that experienced in an undisturbed ionosphere (Figure 3). In particular, the focusing effects have been strengthened and the position of strongest signal considerably altered (note that the structure has caused more than one center of focusing for the first hop). The focusing strongly depends on the phase offset of the wave structure, as can be seen from Figure 4. Since the wave structure will normally be traveling (i.e. the phase offset varies with time), it can be seen that the strength of the received signal at any location will also vary with time (i.e. the fading caused by ionospheric focusing).

### 6. Discussion

[14] Huygen's principle, in the form of integral equations, has the potential to provide a general approach to propagation problems. Complex propagation can be analyzed by dividing the space between transmitter and receiver into a number of regions and advancing the solution between adjacent regions by means of integral equations with simple approximate kernels. In the current paper, it has been demonstrated that the approach can provide useful results when refraction is the major propagation mechanism. In particular, the approach leads to an economic form of the WKB solution. Since the approach breaks the propagation into smaller parts, the method is ideal for analyzing problems in which the nature of propagation is substantially different over a limited region. In this case, the solutions either side of this special region can be connected by integral equations over the region boundaries. In the case of propagation through a region of material turbulence, this approach could provide an alternative to the more usual phase screen methods [Nickisch, 1992]. Another possible application arises in the study of propagation that involves both refraction and diffraction (ducting problems involving a land sea interface, for example). In this case, point source solutions provided by the WKB method could be coupled with surface integrals through the major sources of diffraction. Such an extension is currently under investigation.

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### Supporting Information

FilenameFormatSizeDescription
rds4890-sup-0001-tab01.txtplain text document1KTab-delimited Table 1.
rds4890-sup-0002-tab02.txtplain text document1KTab-delimited Table 2.

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